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BysVecLinReg

BysVecLinReg is an open source TOL Package published as partt of Official Tol Archive Network

BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with arbitrary constraining inequations and lineal constraining equations.

The method used to solve it in this package is based on ​Bayesian linear regression Thomas Minka (2001) using invariant scale prior over A and inverse prior over V

Vectorial linear regression

Vectorial linear regression equations are

Y=A \cdot X + E

where

  • Y\in\mathbb{R}^{d\times N} is the multivariant known output matrix, where each row is a different output vector y_{n}\in\mathbb{R}^{N}
  • X\in\mathbb{R}^{m\times N} is the known and full rank input matrix, where each row is a different input vector x_{n}\in\mathbb{R}^{N}
  • A\in\mathbb{R}^{d\times m} has the unknown regression coefficients that we want to estimate
  • E\in\mathbb{R}^{d\times N} is the multivariant residuals, where each row is the residuals vector e_{n}\in\mathbb{R}^{N} corresponding to output y_{n}

All residuals inside the same row are incorrelated normal, but resiudals in the same column j are

e_{.,j} \sim N\left(0,V\right) E\in\mathbb{R}^{d\times d} \forall j=1 \ldots d

where V is symmetric positive definite and unknown, but the same for each column.

Minka defines also the known data pair D = left(Y,Xright) that will be used just to get more compact conditioninig expressions.

Arbitrary constraining inequations

We will extend the model scope with arbitrary non null meassured restrictions over parameters inside A by means of adding a set of r inequations defining a feasible region

\Omega = \left\{ A\in\mathbb{R}^{d\times m} \mid F\left(A\right) \le 0 \right\}

being

F\left(A\right):\mathbb{R}^{d\times m}\longrightarrow\mathbb{R}^{r}

the arbitrary constraining function.

Invariant-scale prior over coefficient matrix

Although Minka not explicitly stated in any place, under the invariant prior follows that X must be full-rank m <= N because X W X ^ T must be nonsingular with W = \alpha I_{m}, where \alpha is the scale-invariant parameter governing the prior and estimated more forward to maximize the evidence of the data, which depends on the assumptions the model.

Last modified 14 years ago Last modified on Oct 19, 2010, 10:30:53 AM